Some may think that by making the values for A and B into decimals it would increase rigor in solving the equation for C. While it does make solving harder, especially without a calculator, does it increase the thinking demand on the student?

The answer is no, the thinking demand does not increase. The student still understands that to solve for C they must square the values of A and B, then find the square root of the sum of the resulting products. The only thing that has changed is the amount of time it will take to find the products, add them, and find the square root of their sum.

To increase the thinking demand, we must infuse the problem with critical thinking.

Ex. 3

Jalen spent $37.50 on 30 feet of paving stones to pave a diagonal path from the northwest corner of his rectangular garden to the southeast corner of the garden. The width of the garden is 18 ft. How much would it cost for Jalen to pave the perimeter of the garden with the same paving stones?

In Example 3, students have to analyze the problem. Most likely a middle school students would draw a picture of the garden with the diagonal path. At this point, synthesis of prior knowledge of rectangles, perimeter, and Pythagorean Theorem come together in a meaningful way to solve a rigorous problem. They would remember that rectangles have 90 degree angles, thus the diagonal represents the hypotenuse of a right triangle. From here, they realize that since they are given the length of the hypotenuse and a leg, that they are solving for the other leg which is needed to determine the perimeter. Once the student has found the perimeter, they will find the cost of the paving stone per foot, and use the unit rate to determine the cost of paving the perimeter of the garden with the same paving stone.

To extend this problem further, and incorporate the "Create" thinking level, you could add part B..."Suppose Jalen wanted to create a garden that had a diagonal path which was twice the length of his original path. What would be the perimeter of the new garden? Is there only one solution? Why or why not?"

I have seen many teachers force students into memorizing Pythagorean Theorem, and drill students with "naked" math problems for practice. (Naked math problems are math problems that are not in context. They are numbers, equations, expressions with no contextual framing). This is at best, the thinking level of "Apply" in Bloom's Taxonomy. In today's educational system, whether you are a Common Core State or not, accountability tests no longer offer the low rigor problems without context. So my question to teachers who are still teaching naked math is, "What are you preparing your students for?" The same level of practice can be achieved by using problem in context.

In Texas, the state accountability test is the STAAR, State of Texas Assessments of Academic Readiness. The STAAR Mathematics test is comprised of 100% word problems, of which the majority require analysis and synthesis to negotiate the multiple steps to find solutions to problems in almost every subject, especially mathematics. With this is mind, it is my belief that classroom instruction of mathematics should be rigorous and complex, providing opportunities for students to practice solving problems in context that require higher levels of thinking.

Rigor in education doesn't simply mean harder problems on worksheets or more problems in a set. Rigor means the level of thinking required to solve a problem. In other words, a rigorous lesson includes opportunities for students to analyze, synthesize and create. A simple example would be a lesson on Pythagorean Theorem. We could teach students the algorithm...